Details

Autor(en) / Beteiligte

• Dembo, Amir
• Tsai, Li-Cheng

Titel

EQUILIBRIUM FLUCTUATION OF THE ATLAS MODEL

Ist Teil von

• The Annals of probability, 2017-11, Vol.45 (6B), p.4529-4560

Ort / Verlag

Institute of Mathematical Statistics

Erscheinungsjahr

2017

Quelle

Project Euclid Complete

Beschreibungen/Notizen

• We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (ℤ+-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on ℝ+. In this context, we show that the joint law of ranked particles, after being centered and scaled by t−¼, converges as t → ∞ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on ℝ+ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.